Noncut subsets of the hyperspace of subcontinua

doi:10.1016/j.topol.2021.107867

Topology and its Applications

Volume 305, 1 January 2022, 107867

https://doi.org/10.1016/j.topol.2021.107867 Get rights and content

Abstract

Let $C (X)$ be the hyperspace of all subcontinua of a continuum X topologized by the Hausdorff metric. For a nonempty closed subset A of a continuum X, consider the following properties: A is subset of colocal connectedness, strong noncut subset, weak nonblock subset, shore subset, not strong center and noncut subset of X. We study the relation between a one-point subset ${p}$ of X having one of these properties in X and the subset $C_{p} (X) = {B \in C (X) : p \in B}$ and maximal order arcs α from ${p}$ having the same property in $C (X)$ .

Introduction

A continuum is a compact, connected, nonempty, metric space. The hyperspace of all nonempty closed subsets of a continuum X is denoted by $2^{X}$ , the symbol $C (X)$ represents the hyperspace of all subcontinua of X and the hyperspace of all singletons of X is denoted by $F_{1} (X)$ . These hyperspaces are considered with the Hausdorff metric H (see [22, p. 1]).

In continuum theory, there are different concepts related to the idea of being on the “edge” of a continuum or being a sort of nuncut subset of a continuum. In [21] the authors study the following concepts in the context of hyperspaces of continua.

Let X be a continuum and let $B \in 2^{X}$ such that ${int}_{X} (B) = \emptyset$ . We say that B is:

•
a subset of colocal connectedness of X provided that for each open subset U of X that contains B there exists an open subset V of X such that $B \subseteq V \subseteq U$ and $X ∖ V$ is connected.
•
a strong noncut subset of X provided that $X ∖ B$ is continuumwise connected, i.e., for any two points $x, y \in X ∖ B$ there exists a subcontinuum A of X such that $x, y \in A \subseteq X ∖ B$ .
•
a nonblock subset of X provided that $⋃ {L \in C (X) : q \in L \subseteq X ∖ B}$ is dense in X for every $q \in X ∖ B$ .
•
a weak nonblock subset of X provided that $⋃ {L \in C (X) : q \in L \subseteq X ∖ B}$ is dense in X for some $q \in X ∖ B$ .
•
a shore subset of X provided that for each $ε > 0$ there is a subcontinuum M of X such that $H (M, X) < ε$ and $M \cap B = \emptyset$ .
•
not a strong center of X provided that for each pair of nonempty open subsets U and V of X there exists a subcontinuum M of X such that $M \cap U \neq \emptyset \neq M \cap V$ and $M \cap B = \emptyset$ .
•
a noncut subset of X if $X ∖ B$ is connected.

For a continuum X, let

•
$C L C (X) = {B \in 2^{X} : B is a subset of colocal connectedness of X}$ ;
•
$S N C (X) = {B \in 2^{X} : B is a strong noncut subset of X}$ ;
•
$N B (X) = {B \in 2^{X} : B is a nonblock subset of X}$ ;
•
$N B^{⁎} (X) = {B \in 2^{X} : B is a weak nonblock subset of X}$ ;
•
$S (X) = {B \in 2^{X} : B is a shore subset of X}$ ;
•
$N S C (X) = {B \in 2^{X} : B is not a strong center of X}$ ; and
•
$N C (X) = {B \in 2^{X} : B is a noncut subset of X}$ .

These subsets of

2^{X}

are called hyperspaces of noncut sets of X. Readers specially interested in the study of these hyperspaces are referred to [1], [2], [3], [4], [5], [8], [10], [12], [15], [16], [19], [20], [21], [24], [27].

The aim of [21] is to prove that $F_{1} (X)$ belongs to some hyperspace of noncut sets of the hyperspaces $2^{X}$ , $C_{n} (X) = {A \in 2^{X} : A has at most n components}$ and $F_{n} (X) = {A \in 2^{X} : A has at most n elements}$ . In this paper, we are interested in studying all possible relationships among the conditions:

(a)
${p}$ is an element of certain hyperspace of noncut sets $H (X)$ of a continuum X;
(b)
$C_{p} (X) = {B \in C (X) : p \in B}$ is an element of $H (C (X))$ ; and
(c)
an order arc from ${p}$ to X is an element of $H (C (X))$ .

More precisely, we present results related to the relationships between (a) and (b), and between (a) and (c).

Section snippets

Preliminaries and auxiliary results

Let X be a continuum. The interior and the closure of a subset A of X are denoted by ${Int}_{X} (A)$ and ${Cl}_{X} (A)$ , respectively. The open ball of radius $ε > 0$ and center $x \in X$ is denoted by $B (ε, x)$ and the symbol $N (A, δ)$ denotes the set $⋃ {B (δ, a) : a \in A}$ for each subset A of X and for each $δ > 0$ . The Hausdorff metric for $C (X)$ is denoted by H and $H_{2}$ represents the Hausdorff metric for $C (C (X))$ induced by H.

Theorem 2.1

For a continuum X, the following inclusions hold: $C L C (X) \subseteq S N C (X) \subseteq N B (X) \subseteq N B^{⁎} (X) \subseteq S (X) \subseteq N S C (X) \subseteq N C (X) .$

Proof

In light of [8,

The containment hyperspaces $C_{p} (X)$

Let X be a continuum and let $μ : C (X) \to [0, 1]$ be a Whitney map. A nonempty closed subset of $C (X)$ of the form $μ^{- 1} ([r, s])$ is called a Whitney block of X. From [17, Theorem 19.9, p. 160] and [30, (2.2) Chapter VIII, p. 138] together, it follows that each Whitney block is a subcontinuum of $C (X)$ . For a point p of X and a Whitney block $W$ of X, the set $W \cap C_{p} (X)$ is denoted by $W_{p}$ and the set $μ^{- 1} (t) \cap C_{p} (X)$ by $μ_{p}^{- 1} (t)$ .

Lemma 3.1

Let X be a continuum, let $p \in X$ and let $W$ be a Whitney block of X. If $U$ is an open subset of $W$

Maximal order arcs

An arc is any space homeomorphic to $[0, 1]$ . An order arc α in the hyperspace of subcontinua of $C (X)$ of a continuum X is an arc contained in $C (X)$ such that if $A, B \in α$ , then either $A \subseteq B$ or $B \subseteq A$ . The existence of order arcs in $C (X)$ is guaranteed by [17, Theorem 14.6, p. 112]

Let X be a continuum. For a point $p \in X$ , we denote by $Λ (X, p)$ the set of all order arcs from ${p}$ to X. Define $Λ (X) = ⋃_{p \in X} Λ (X, p)$ . The set $Λ (X)$ is called the space of maximal order arcs in $C (X)$ and is topologized as subspace of $C (C (X))$ .

Let

Acknowledgement

We are thankful to the referee for her/his comments that improved the presentation of the paper.

References (30)

J. Bobok et al.
Union of shore sets in a dendroid
Topol. Appl.
(2014)
J. Bobok et al.
On blockers in continua
Topol. Appl.
(2016)
J. Camargo et al.
Continua whose hyperspace of nonblockers of $F_{1} (X)$ is a continuum
Topol. Appl.
(2019)
J. Camargo et al.
The hyperspace of nonblockers of $F_{1} (X)$
Topol. Appl.
(2019)
R. Escobedo et al.
On hyperspaces of non-cut sets of continua
Topol. Appl.
(2017)
A. Illanes
The cone = hyperspace property, a characterization
Topol. Appl.
(2001)
A. Illanes et al.
Blockers in hyperspace
Topol. Appl.
(2011)
R. Leonel
Shore points of a continuum
Topol. Appl.
(2014)
V.C. Nall
Centers and shore points of a dendroid
Topol. Appl.
(2007)
C. Piceno
Nonblockers in homogeneous continua
Topol. Appl.
(2018)

J. Bobok et al.

Non-cut, shore and non-block points in continua

Glas. Mat. Ser. III

(2016)

J.J. Charatonik

Generalized homogeneity and some characterizations of solenoids

Bull. Pol. Acad. Sci., Math.

(1983)

D.W. Curtis

Application of a selection theorem to hyperspace contractibility

Can. J. Math.

(1985)

J.T. Goodykoontz

More on connectedness im kleinen and local connectedness in $C (X)$

Proc. Am. Math. Soc.

(1977)

E.E. Grace

Aposyndesis and Weak Cutting, General Topology and Modern Analysis

(1981)

Cited by (0)

¹: The work was supported by the grant UAEM 4977/2020CIC.

View full text

Noncut subsets of the hyperspace of subcontinua

Abstract

Introduction

Section snippets

Preliminaries and auxiliary results

The containment hyperspaces Cp(X)

Maximal order arcs

Acknowledgement

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Topol. Appl.

Non-cut, shore and non-block points in continua

Glas. Mat. Ser. III

Generalized homogeneity and some characterizations of solenoids

Bull. Pol. Acad. Sci., Math.

Application of a selection theorem to hyperspace contractibility

Can. J. Math.

More on connectedness im kleinen and local connectedness in C(X)

Proc. Am. Math. Soc.

Aposyndesis and Weak Cutting, General Topology and Modern Analysis

The containment hyperspaces $C_{p} (X)$

More on connectedness im kleinen and local connectedness in $C (X)$